Maximally Flat FIR digital filters are frequency-selective discrete-time systems also known as smooth filters. They are thought to be important because of the monotone and flat magnitude response that many of them exhibit. Researchers have used the half-band and linear-phase members of the family to design the so-called regular wavelets. There are other reasons to study this class of digital filters. We have shown in an our recent paper that the lowpass maximally flat filters are suitable for processing polynomial signals. The interested reader is encouraged to read our paper in ISCAS'2001.

In most of the contemporary approaches to maximally flat designs,
the **optimality criterion** is the assignment of all degrees of freedom to the following two objectives.

- To guarantee the desired magnitude responses at specified frequencies called
**centers of flatness.** - To ensure that a maximum possible number of derivative coefficients of the magnitude response are strictly equal to zero at the centers of flatness.

The problem of design of such magnitude response characteristics
is equivalent to solving a form of *Hermite Interpolation* problem. Hermite
interpolation is known to possess unique solutions when solved in the space of univariate polynomials.

This article is mainly concerned with **1-D maximally flat
filters**. Needless to say, like all other 1-D filter design
criteria, maximally flat designs can be extended to the two, and
generally, the **multi-dimensional (M-D) filters**. For
M-D filters, the design is usually problematic because the M-D Hermite
interpolation problem is not always solvable. In fact the problem of
multivariate Hermite interpolation is still an active research area in
mathematics. M-D maximally flat filters are thought to be useful for
band limitation in sampling rate conversion of TV signals and other M-D
digital waveforms.

Here we give a list of some main characteristics of maximally flat lowpass and highpass filters that often appear in literature.

Magnitude Response |
Flat around the centers of flatness. Usually monotone. It is possible to design filters with lowpass, highpass, bandpass, bandstop or even multi-band amplitude responses. |

Phase Response |
Both linear phase and nonsymmetric (nonlinear-phase) filters exist. However, even for the nonsymmetric filters, the phase response is always flat around the center of flatness. |

Impulse Response |
Duration of the impulse response is finite. It assumes
rational values for all linear-phase filters and those nonlinear-phase filters
that have a rational group delay at w=0. |

Cutoff Frequency |
For linear phase filters under a fixed impulse response length equal to N, it is only possible to realize (N-1)/2 different cutoff frequencies. For nonlinear-phase lowpass and highpass filters, it is possible to adjust the cutoff frequency at a desirable value by varying the amount of group delay at the center of flatness. |

Transition Band |
Compared to filters that are designed according to minimax or least squares criteria, the transition band is wider. Furthermore, the improvement in the transition band width caused by an increase in the length of the filter is sluggish. Specifically, for linear-phase filters the order for the filter increases as the second power of the inverse of the required transition width. It seems that for applications requiring very narrow transition characteristics, maximally flat filters should be thought of as the least favorable option. |

Perhaps the most intriguing aspect of maximally flat filters is that closed-form solutions exist in most situations. Another well-known case where closed-form formulas are available for optimum FIR filters is the window-based filters that are optimal in the least squares sense.

The well-known closed-form solutions, employed by several researchers in their papers, are related to the linear-phase case. However, the linear-phase solutions are only the tip of iceberg. They are simply a special case of the large class of ``universal maximally flat lowpass FIR filters.'' The class contains linear-phase as well as nonsymmetric filters. All the members of the class enjoy closed-form transfer functions. These filters were discovered by Baher in 1982. They remained mainly unrecognized until recent years. In a recent paper we provided an alternative formula for the filters, proved their universality and developed array structures for realization of some special cases.

The formulas are listed in the following section.

- Linear phase filters (type 1)
- General Lowpass and highpass filters (including half-band filters)
- Herrmann's Formula
- Miller's Formula (Bernstein Form)
- Fahmy's Formula (Integral Representation)
- Hamming's Formula (Integral Representation)

- Half-band filters (type 1)
- Gumacos' Formula
- Our Formula [From Samadi et al., IEEE CAS II, 1999]

- Herrmann's Formula

- General Lowpass and highpass filters (including half-band filters)
- Nonsymmetric Filters
- Baher's Formula: We have proved that this is one of the most general formulas known to this date. We used an automatic method for proving identities that comprise sum of hypergeometric terms. The method is originally based on the work of Sister Mary Celine Fasenmyer. It was mainly developed by D. Zeilberger and H. Wilf. Useful software packages for use with computer algebra systems are developed by Professor Peter Paule at Linz. Here is a pdf file of the paper.
- Generalized Half-Band Filters: For more information see this article.

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